Be sure to check the home page for roles listed in the rubric. Also, please read the home page for information and updates.
Text (Notes) - Amanda Saravia and Gabby Sallada Practice Problems - Cheyenne Stambaugh and Mary Sweitzer
Real-Life Applications- Devin Taylor
Multimedia - Alex schmitt
SECTION 6.1 Properties of Exponents
Name Of Property
Algebra
Example
*Product Of Powers*
a^m x a^n = a^m+n
2^3 x 2^4 = 27
Power Of A Power
(a^m)^n = a^mn
(2^4)^2 = 2^8
Power Of A Product
(ab)^m = a^m x b^m
( 3 x 4)^2 = 3^2 x 4^2
Negative Exponent
a^-m= 1/a^m
4^-2 = 1/4^2
Zero Exponent
a^0 = 1, a ≠ 0
4^0 = 1
*Quotient Of Powers*
a^m/a^n = a^m-n
5^2/5^1 = 5^1
Power Of A Quotient
(a/b)^m = (a^m/b^m)
(6/9)^2 = (6^2/9^2)
*Stars mean that bases need to be the same.*
Scientific Notation A number expressed in the form c x 10^n where c is greater than or equal to one and less than ten and n is an integer.Example: .4500
4.5 x 10^3
PRACTICE PROBLEMS Evaluate the following expressions. Name the property that is used.
1. (2^3)^4
2. (3/4)^2
3. (-5)^-6(-5)^4
4. x^0
5. (7b^-3)^2b^5b
Word Problem:
An adult human body contains about 75,000,000,000,000 cells. Each cell is about 0.001 inch wide. If the cells were laid end to end to form a long chain, about how long would the chain be in miles? Give the answer in scientific notation.
Real-Life Applications- Properties of Exponents can be used to find the ratio of a state's park space to total area Section 6.2
Polynomials-a sum of terms where each term is a constant times a whole number power of x. Standard Form -when in standard form equation will have highest exponent of variable first and others will be in decreasing order.
-Monomial- 1 term
-Binomial-2 terms -Trinomial-3 terms
Direct Substitution-when sustituting the x value in for x in order to sove the equation Steps
1.) substitute x value in for x in the equation 2.) solve equation
ex.4x^4+3x^2+2x-1, x=4 4(4)^4=3(4)^2+2(4)-1
4(256)+3(16)+8-1 1024+48+7
1079
Synthetic Substitution-will be using coefficents and a given value for x Steps 1.) write the value of x and the coefficents from the equation in decreasing order from left to right
2.) bring down the leading coefficent and multiply it by the x value 3.) write the answer in the next column and add the numbers in that column 4.) write the sum below the line and multiply it by the x value and put the product in the next colmn
5.) continue to add and multiply until the last coefficent has been added to a number and that gives you your answer ex. 4x^4+3x^2+2x-1 4------0------3------2------(-1) -----(16)---(64)--(268)--(1080)
(4)--(16)--(67)--(270)-(1079)
final answer: 1079 2-----0----(-8)-----5-----(-7)
-----(6)---(18)---(30)--(105)
2---(6)---(10)--(35)----(98)
final answer: 98
End Behavior of Polynomial Functions-end behavior of a polynomial function's graph is the behavior of a graph as x approches positive infinity(+oo) or negative infinity(-oo) f(x) is y up to +oo(infinity) or down to -oo
x is positive +oo(right) or negative -oo(left)
ex. Put Graph here
ex. Put different graph, unlike first one,
1.) First, use a table of values to get started-> x I y
2.) Then,plug the different x values from the -3 I ? table of values into the equation and put -2 I ?
the different answers into the table ofvalues. -1 I ?
3.) Lastly, plot the points from thetable of values 0 I ?
onto the graph 1 I ?
2 I ?
3 I ? .
ex. PUT GRAPH HERE FOR EQUATIONS F(X)=4X+2 AND F(X)= -X^4-2X^3+2X^2+4X
Practice Problems for 6.2State whether or not the given function is a polynomial. If so, then state the degree, type, and leading coefficient.
1. f(x)= 12-5x
2. f(x)= x+π
3. f(x)= -2x^3-3x^-3+x
4. f(x)= x^2-x+1
5. f(x)= 3x^3
When x=5, evaluate the given polynomial functions using direct substitution.
1. f(x)= 2x^3+5x^2+4x+8
2. f(x)=x+1/2x^3
3. f(x)=5x^4-8x^3+7x^2
4. f(x)=11x^3-6x^2+2
5. f(x)=7x^3+9x^2+3x
When x=-3, evaluate the following fuctions using synthetic division.
1.f(x)= 3x^5+5x^3+2x+9
2. f(x)=x^2+1/2x-3
3. f(x)=x^4-6x^2+7x
4. f(x)=4x^3+x^2+2
5. f(x)=7x^4+5x^3+3x
Graph each function, then describe its end behavior.
1.-5x^3-2
2.x^2+1
3.x^5-2x^2+2
Real-Life Application- Evaluating and Graphing can be used in real life for example by finding out how much money is awarded at the U.S. open. Combining Polynomials AddingAdd like terms-varibles and exponents are the same----->ex. (3x^2+7+x)+(14x^3+2+x^2-X)=14x^3+4x^2+9-->Adding horizontially
-----------standard form-> 3x^2+x+2 adding vertically->+14x^3+x^2-x+2answer------>14x^3+4x^2+9
Subtracting8x^3-3x^2-2x+9 -2x^3-6x^2+x-1
8x^3-3x^2-2x+9 -(2x^3+6x^2-x+1)<-----add the opposite 6x^3-9x^2-x+8
(2x^2+3x)-(3x^2+x-4)=2x^2+3x-3x^2-x+4=-x^2+2x+4
Multiplication When you multiply, all you do is foil the problem out and then combine like terms and put in standard form. ex. (4x - 5) (2x^5 + x^3 -1)
FOIL->8x^6 + 4x^4 - 4x - 10x^5 - 5x^3 + 5
8x^6 - 10x^5 + 4x^4 - 5x^3 - 4x +5
*Box Method*
(3x-4)(8x-1) l 3x l -4
8x l 24x^2 l -32x
-1 l -3x l 4
then, add like terms and you get 24x^2-35x+4 *when multiplying any kind of 2 polynomials box method can be used* Long Division
First put the problem into standard form. Then divide like you would divide a regular problem. If you are left with a remainder or not zero put that over the divisor. 2x_+_1_+_5 /x+4
x + 4 ) 2x^2 + 9x + 9 - 2x^2 - 8x
x +9 -x - 4
5
Synthetic Division
The number in the box should make the linear binomial equal zero. It is an intercept. The boxed number is a remainder. To do synthetic division you do the same as synthetic substitution.
Real-Life Applications- Combining polynomials can be used in real life by seeing how much power is needed to keep a bike moving at a certain speed. Theorems Remainder Theorem - If a polynomial f(x) is divided by ( x - k ), then the remainder is f(k) = r. In other words, when you divide using synthetic division, such as below, the remainder will bethe same as when you use direct substitution. ex. ( 3x^2 + 5x - 2) / ( x + 4)
4] 3 5 -2 -12 28
3 -7 26
3(-4^2) + 5(-4) - 2
3(16) - 20 - 2
48 - 2 - 2 26
Factor Theorem- A polynomial f(x) has a factor ( x - k ) if and only if f(k) = 0. ex. x^2-4
(x+2)(x-2)
-2] 1 0 -4 --------(-2 4
-----(1 -2[0
2] 1 0 -4
-------(2 4
----(1 2[0
-2 and 2 are the "zeros" of the polynomial. they also are the x intercepts
ex. 2x^3+5x^2-x+7 when x=2
2(2)^3+5(2)^2-2+7
32+20+5=41
2] 2 5 -1 7
-------(4 18 34
---(2 9 17[41
Rational Zero Theorem- p/q = factor of constant term a0 / factor of leading coefficient
To find the zero use the p/q test to find all possible zeros. Then use synthetic division to see if one of the possible zeros is a zero. Next the equation you get from synthetic division needs to be factored. Once its factored you can solve for the rest of the zeros.
Fundamental Theorem - If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.
Imaginary roots always come in pairs.
Real-Life Applications- In factoring and solving is finding the dimensions of a block discovered underwater.
Section 6.8
Let f(x) = anx^n + an - 1x^n-1 + ... + a1x + a0. The following statements are equivalent.
Zero - k is zero of the polynomial function f
Factor - x - k is a factor of the polynomial f(x)
Solution - k is a solution of the polynomial equation f(x) = 0.
If k is a real number, then the following is a also equivalent.
X intercept- k is an x-intercept of the graph of the polynomial function f.
Local Maximum - the y coordinate of a turning point that is higher than all nearby points.
Local Minimum - the y coordinate of a turning point that is lower than all nearby points.
Turning Points of Polynomial Functions
The graph of every polynomial function of degree n has at most n - 1 turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly n - 1 turning points.
Practice Problems for 6.3:
#1.Find the sum or difference.
a.(-3x^3+x-11) - (4x^3+x^2-x)
b.(10x^3-4x^2+3x) - (x^3-x^2+1)
c.(10x-3+7x^2) + (x^3-2x+17)
#2. Multiply the polynomials.
a.x(x^2+6x-7)
b.-4x(x^2-8x+3)
c.(x-4)(x-7)
#3. Multiply the three binomials.
a.(x+9)(x-2)(x-7)
b.(x+5)(x+7)(-x+1)
c.(x-9)(x-2)(3x+2)**
Practice Problems for 6.4:
Factor the polynomial using any method.
1.x^6+125
2.X^4-1
3.5x^3-320
4.3x^2+11x+6
5.125x^3-216
Find the GCF
1.3x^4-12x^3
2.24x^4-x^6
3.145x^9-17
Factor.
1. x^3-8
2. 216x^3+1
3. 1000x^3+27
Factor by Grouping
1. x^3+x^2+x+1
2.x^3+3x^2+10x+30
3.2^3-5x^2+18x-45
Find the real number solutions.
1.2x^3-6x^2=0
2.x^3+27=0
3.x^4+7X^3-8x-56=0
4.3X^7-243x^3=0
5.8x^3-1=0 Real-Life Applications- In Factoring and Solving polynomial equations is finding the dimensions of a block discovered underwater
Practice Problems for 6.5:
#1. Use Synthetic Division.
a. Dividex^3+2x^2-6x-9 by x+3
b. Divide2x^4+3x^3+5x-1 by x-2 c. Divide 4x^3-x^2+5x-1 by x-4
#2.Find the factors of the polynomial given that f(k)=0.
a.f(x)=2x^3+11x^2+18x+9 k= -3
b.f(x)=x^3-5x^2-2x+24 k= -2
c.f(x)=4x^3-4x^2-9x+9 k=1
#3.Find the other zeros of the polynomial function.
a. One zero of f(x)=x^3-2x^2-9x+18 is x=2
b. One zero of f(x)=9x^3+10x^2-17x-2 is x= -2
c. One zero of f(x)=2x^3+3x^2-39x-20 is x=4
#4. Use polynomial long divison.
a.(x^2+7x-5) / (x-2)
b.(2x^2+3x-1) / (x+4)
c.(x^2+5x-3) / (x-10)
Real-Life Applications- In in the Remainder an Factor Theorems is used in finding a production level that yields a certain profit
Practice Problems for 6.6:
#1.Find all possible rational zeros.
a.f(x)=x^3+2x^2-11x-12
b.f(x)=x^4+2x^2-24
c.f(x)=2x^5+x^2+16
#2. Using sythetic divison decide which of the following are zeros of the function; 1, -1, 2, -2.
a.f(x)=x^3+7x^2-4x-28
b.f(x)=x^4+3x^3-7x^2-27x-18
c.f(x)=x^4+3x^3+3x^2-3x-4
#3.Find all the real zeros of the function.
a.f(x)=x^3-8x^2-23x+30
b.f(x)=x^3-7x^2+2x+40
c.f(x)=x^3+72-5x^2-18x
Real-Life Applications- In Finding Rational Zeros can be used to find the dimensions of a monument
6.7 Real-Life Applications- In Fundamental Theorem is used in real life by finding the population of certain people
Practice Problems for 6.8:
#1. Estimate the coordinates of each turning point and state whether each cooresponds to a local maximum or a local minimum. Then list all the real zeros and determine the least degree that the function can have.
do numbers 23, 25 and 27.
#2.Use a graphing calculator to graph the polynomial function. Identify the x-intercepts and the points where te local maximums and local minimums occur.
a.f(x)=3x^3-9x+1
b.f(x)= -1/4x^4+2x^2
c.f(x)=x^5-5x^3+4x
#3. Graph the function.
a.f(x)=(x-1)^3(x+1)
b.f(x)=1/8(x+4)(x+2)(x-3)
c.f(x)=5(x-1)(x-2)(x-3)
Real-Life Applications- In Analyzing graphs is used to find the certain crop productions in the United States
Answers to practice problems
Be sure to check the home page for roles listed in the rubric. Also, please read the home page for information and updates.Text (Notes) - Amanda Saravia and Gabby Sallada
Practice Problems - Cheyenne Stambaugh and Mary Sweitzer
Real-Life Applications- Devin Taylor
Multimedia - Alex schmitt
SECTION 6.1
Properties of Exponents
Scientific Notation
A number expressed in the form c x 10^n where c is greater than or equal to one and less than ten and n is an integer.Example: .4500
4.5 x 10^3
PRACTICE PROBLEMS
Evaluate the following expressions. Name the property that is used.
1. (2^3)^4
2. (3/4)^2
3. (-5)^-6(-5)^4
4. x^0
5. (7b^-3)^2b^5b
Word Problem:
An adult human body contains about 75,000,000,000,000 cells. Each cell is about 0.001 inch wide. If the cells were laid end to end to form a long chain, about how long would the chain be in miles? Give the answer in scientific notation.
Real-Life Applications- Properties of Exponents can be used to find the ratio of a state's park space to total area
Section 6.2
Polynomials-a sum of terms where each term is a constant times a whole number power of x.
Standard Form -when in standard form equation will have highest exponent of variable first and others will be in decreasing order.
Polynomials are classified by degrees.
Name----------------Degree-----------------Example
Constant-------------- 0 ---------------------- 4x^0=4(1)=4
Linear------------------ 1 ----------------------- 1/2x+3
Quadratic------------- 2 ----------------------- 4x^2+3x+2
Cubic------------------ 3 ---------------------- 5x^3+4x^2+x+3
Quartic---------------- 4 ----------------------- 8x^4+2x^3+x^2+x+3
Quintic---------------- 5 ----------------------- x^5-4
Leading coefficent is the number in front of a value with the greatest exponent
ex. -3x^4+1/2x^2-7 leading coefficent is -3
*Constant*=highest exoponent
*exponents must be positive integers*
*Polynomial functions
ex. f(x)= 5x+7 ex. f(x)= x^2+4x-7 ex. 3/2x^4-2/3x^2
*Not Polynomial functions
ex. f(x)= 2x+3/5x-2 ex. 2x^-3 ex. f(x)= 1/x^2-5x+3
-Monomial- 1 term
-Binomial- 2 terms
-Trinomial- 3 terms
Direct Substitution-when sustituting the x value in for x in order to sove the equation
Steps
1.) substitute x value in for x in the equation
2.) solve equation
ex.4x^4+3x^2+2x-1, x=4
4(4)^4=3(4)^2+2(4)-1
4(256)+3(16)+8-1
1024+48+7
1079
Synthetic Substitution-will be using coefficents and a given value for x
Steps
1.) write the value of x and the coefficents from the equation in decreasing order from left to right
2.) bring down the leading coefficent and multiply it by the x value
3.) write the answer in the next column and add the numbers in that column
4.) write the sum below the line and multiply it by the x value and put the product in the next colmn
5.) continue to add and multiply until the last coefficent has been added to a number and that gives you your answer
ex. 4x^4+3x^2+2x-1
4------0------3------2------(-1)
-----(16)---(64)--(268)--(1080)
(4)--(16)--(67)--(270)-(1079)
final answer: 1079
2-----0----(-8)-----5-----(-7)
-----(6)---(18)---(30)--(105)
2---(6)---(10)--(35)----(98)
final answer: 98
End Behavior of Polynomial Functions- end behavior of a polynomial function's graph is the behavior of a graph as x approches positive infinity(+oo) or negative infinity(-oo)
f(x) is y up to +oo(infinity) or down to -oo
x is positive +oo(right) or negative -oo(left)
ex. Put Graph here
ex. Put different graph, unlike first one,
Graphing Polynomials
1.) First, use a table of values to get started-> x I y2.) Then,plug the different x values from the -3 I ?
table of values into the equation and put -2 I ?
the different answers into the table ofvalues. -1 I ?
3.) Lastly, plot the points from thetable of values 0 I ?
onto the graph 1 I ?
2 I ?
3 I ?
.
ex. PUT GRAPH HERE FOR EQUATIONS F(X)=4X+2 AND F(X)= -X^4-2X^3+2X^2+4X
Practice Problems for 6.2State whether or not the given function is a polynomial. If so, then state the degree, type, and leading coefficient.
1. f(x)= 12-5x
2. f(x)= x+π
3. f(x)= -2x^3-3x^-3+x
4. f(x)= x^2-x+1
5. f(x)= 3x^3
When x=5, evaluate the given polynomial functions using direct substitution.
1. f(x)= 2x^3+5x^2+4x+8
2. f(x)=x+1/2x^3
3. f(x)=5x^4-8x^3+7x^2
4. f(x)=11x^3-6x^2+2
5. f(x)=7x^3+9x^2+3x
When x=-3, evaluate the following fuctions using synthetic division.
1.f(x)= 3x^5+5x^3+2x+9
2. f(x)=x^2+1/2x-3
3. f(x)=x^4-6x^2+7x
4. f(x)=4x^3+x^2+2
5. f(x)=7x^4+5x^3+3x
Graph each function, then describe its end behavior.
1.-5x^3-2
2.x^2+1
3.x^5-2x^2+2
Real-Life Application- Evaluating and Graphing can be used in real life for example by finding out how much money is awarded at the U.S. open.
Combining Polynomials
AddingAdd like terms-varibles and exponents are the same----->ex. (3x^2+7+x)+(14x^3+2+x^2-X)=14x^3+4x^2+9-->Adding horizontially
-----------standard form-> 3x^2+x+2
adding vertically->+14x^3+x^2-x+2 answer------>14x^3+4x^2+9
Subtracting8x^3-3x^2-2x+9
-2x^3-6x^2+x-1
8x^3-3x^2-2x+9
-(2x^3+6x^2-x+1)<-----add the opposite
6x^3-9x^2-x+8
(2x^2+3x)-(3x^2+x-4)=2x^2+3x-3x^2-x+4=-x^2+2x+4
Multiplication
When you multiply, all you do is foil the problem out and then combine like terms and put in standard form.
ex. (4x - 5) (2x^5 + x^3 -1)
FOIL->8x^6 + 4x^4 - 4x - 10x^5 - 5x^3 + 5
8x^6 - 10x^5 + 4x^4 - 5x^3 - 4x +5
*Box Method*
(3x-4)(8x-1)
l 3x l -4
8x l 24x^2 l -32x
-1 l -3x l 4
then, add like terms and you get 24x^2-35x+4
*when multiplying any kind of 2 polynomials box method can be used*
Long Division
First put the problem into standard form. Then divide like you would divide a regular problem. If you are left with a remainder or not zero put that over the divisor.
2x_+_1_+_5 /x+4
x + 4 ) 2x^2 + 9x + 9
- 2x^2 - 8x
x +9
-x - 4
5
Synthetic Division
The number in the box should make the linear binomial equal zero. It is an intercept. The boxed number is a remainder. To do synthetic division you do the same as synthetic substitution.
ex. -3] 6 -5 -6
------------(-18 69
---------(6 -23 [63]
6x - 23 + 63 /x + 3
Real-Life Applications- Combining polynomials can be used in real life by seeing how much power is needed to keep a bike moving at a certain speed.
Theorems
Remainder Theorem - If a polynomial f(x) is divided by ( x - k ), then the remainder is f(k) = r.
In other words, when you divide using synthetic division, such as below, the remainder will bethe same as when you use direct substitution.
ex. ( 3x^2 + 5x - 2) / ( x + 4)
4] 3 5 -2
-12 28
3 -7 26
3(-4^2) + 5(-4) - 2
3(16) - 20 - 2
48 - 2 - 2
26
Factor Theorem- A polynomial f(x) has a factor ( x - k ) if and only if f(k) = 0.
ex. x^2-4
(x+2)(x-2)
-2] 1 0 -4
--------(-2 4
-----(1 -2 [0
2] 1 0 -4
-------(2 4
----(1 2 [0
-2 and 2 are the "zeros" of the polynomial. they also are the x intercepts
ex. 2x^3+5x^2-x+7 when x=2
2(2)^3+5(2)^2-2+7
32+20+5=41
2] 2 5 -1 7
-------(4 18 34
---(2 9 17 [41
Rational Zero Theorem- p/q = factor of constant term a0 / factor of leading coefficient
To find the zero use the p/q test to find all possible zeros. Then use synthetic division to see if one of the possible zeros is a zero. Next the equation you get from synthetic division needs to be factored. Once its factored you can solve for the rest of the zeros.
ex. f(x) = x^3 + 2x^2 - 11x - 12
p/q = +/- 1, 2, 3, 4, 6, 12
- 1 ] 1 2 -11 -12
-1 -1 12
1 1 -12 0
f(x) = (x +1) (x^2 + x - 12)
f(x) = (x + 1) ( x - 3) ( x + 4)
zeros are -1, 3, and -4
Fundamental Theorem - If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.
Imaginary roots always come in pairs.
Real-Life Applications- In factoring and solving is finding the dimensions of a block discovered underwater.
Section 6.8
Let f(x) = anx^n + an - 1x^n-1 + ... + a1x + a0. The following statements are equivalent.
Zero - k is zero of the polynomial function f
Factor - x - k is a factor of the polynomial f(x)
Solution - k is a solution of the polynomial equation f(x) = 0.
If k is a real number, then the following is a also equivalent.
X intercept- k is an x-intercept of the graph of the polynomial function f.
Local Maximum - the y coordinate of a turning point that is higher than all nearby points.
Local Minimum - the y coordinate of a turning point that is lower than all nearby points.
Turning Points of Polynomial Functions
The graph of every polynomial function of degree n has at most n - 1 turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly n - 1 turning points.
Practice Problems for 6.3:
#1.Find the sum or difference.
a.(-3x^3+x-11) - (4x^3+x^2-x)
b.(10x^3-4x^2+3x) - (x^3-x^2+1)
c.(10x-3+7x^2) + (x^3-2x+17)
#2. Multiply the polynomials.
a.x(x^2+6x-7)
b.-4x(x^2-8x+3)
c.(x-4)(x-7)
#3. Multiply the three binomials.
a.(x+9)(x-2)(x-7)
b.(x+5)(x+7)(-x+1)
c.(x-9)(x-2)(3x+2)**
Practice Problems for 6.4:
Factor the polynomial using any method.
1.x^6+125
2.X^4-1
3.5x^3-320
4.3x^2+11x+6
5.125x^3-216
Find the GCF
1.3x^4-12x^3
2.24x^4-x^6
3.145x^9-17
Factor.
1. x^3-8
2. 216x^3+1
3. 1000x^3+27
Factor by Grouping
1. x^3+x^2+x+1
2.x^3+3x^2+10x+30
3.2^3-5x^2+18x-45
Find the real number solutions.
1.2x^3-6x^2=0
2.x^3+27=0
3.x^4+7X^3-8x-56=0
4.3X^7-243x^3=0
5.8x^3-1=0
Real-Life Applications- In Factoring and Solving polynomial equations is finding the dimensions of a block discovered underwater
Practice Problems for 6.5:
#1. Use Synthetic Division.
a. Dividex^3+2x^2-6x-9 by x+3
b. Divide2x^4+3x^3+5x-1 by x-2
c. Divide 4x^3-x^2+5x-1 by x-4
#2.Find the factors of the polynomial given that f(k)=0.
a.f(x)=2x^3+11x^2+18x+9 k= -3
b.f(x)=x^3-5x^2-2x+24 k= -2
c.f(x)=4x^3-4x^2-9x+9 k=1
#3.Find the other zeros of the polynomial function.
a. One zero of f(x)=x^3-2x^2-9x+18 is x=2
b. One zero of f(x)=9x^3+10x^2-17x-2 is x= -2
c. One zero of f(x)=2x^3+3x^2-39x-20 is x=4
#4. Use polynomial long divison.
a.(x^2+7x-5) / (x-2)
b.(2x^2+3x-1) / (x+4)
c.(x^2+5x-3) / (x-10)
Real-Life Applications- In in the Remainder an Factor Theorems is used in finding a production level that yields a certain profit
Practice Problems for 6.6:
#1.Find all possible rational zeros.
a.f(x)=x^3+2x^2-11x-12
b.f(x)=x^4+2x^2-24
c.f(x)=2x^5+x^2+16
#2. Using sythetic divison decide which of the following are zeros of the function; 1, -1, 2, -2.
a.f(x)=x^3+7x^2-4x-28
b.f(x)=x^4+3x^3-7x^2-27x-18
c.f(x)=x^4+3x^3+3x^2-3x-4
#3.Find all the real zeros of the function.
a.f(x)=x^3-8x^2-23x+30
b.f(x)=x^3-7x^2+2x+40
c.f(x)=x^3+72-5x^2-18x
Real-Life Applications- In Finding Rational Zeros can be used to find the dimensions of a monument
6.7 Real-Life Applications- In Fundamental Theorem is used in real life by finding the population of certain people
Practice Problems for 6.8:
#1. Estimate the coordinates of each turning point and state whether each cooresponds to a local maximum or a local minimum. Then list all the real zeros and determine the least degree that the function can have.
do numbers 23, 25 and 27.
#2.Use a graphing calculator to graph the polynomial function. Identify the x-intercepts and the points where te local maximums and local minimums occur.
a.f(x)=3x^3-9x+1
b.f(x)= -1/4x^4+2x^2
c.f(x)=x^5-5x^3+4x
#3. Graph the function.
a.f(x)=(x-1)^3(x+1)
b.f(x)=1/8(x+4)(x+2)(x-3)
c.f(x)=5(x-1)(x-2)(x-3)
Real-Life Applications- In Analyzing graphs is used to find the certain crop productions in the United States